Today

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Today

• Sequences
• Sums (series)
• Products
• Applications to Computer Science
• Conversion to radix B
• Mathematical induction

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• Original author of the slides
• Vadim Bulitko
• University of Alberta
• http//www.cs.ualberta.ca/bulitko/W04
• Modified by T. Andrew Yang (yang_at_uhcl.edu)

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Sequences

• Set of elements
• Order is important
• Repetitions are allowed
• Possibly infinite
• Each one is called a term
• Index by an integer -- hence countable
• The first one (if exists) is called the initial
  term
• The last one (if exists) is called the final term
• Explicit/general formula specifies the elements

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Examples

• akk/(k1), kgt0
• bi(i-1)/i, igt1
• The same sequence -- different formulae
• Any sequence has an infinite number of
  explicit/general formulae
• How about this sequence
• -1, 1, -1, 1,

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Series

• A series is a sum of a sequence
• Capital Greek letter ? (sigma) is used
• U? ai aL aUiL
• L is the lower limit
• U is the upper limit
• Either one can be infinite

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Examples

• Sum (-1)k
• 8? (-1)k i0
• 1 (-1) 1 (-1)

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Products

• Use capital Greek letter ? (Pi) to denote a
  product U ? ai aL aU iL
• L is the lower limit
• U is the upper limit
• Either one can be infinite

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Examples

• Factorial
• definition
• number of permutations of n elements
• Sample applications
• binomial coefficients (n choose m)
• http//mathworld.wolfram.com/BinomialCoefficient.h
  tml

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Properties

• Distributive law
• Sums
• Products
• Multiplication by a constant

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Applications to CS

• Iterative constructs (loops)
• Example
• for (int IL IltU I)
• Cout ltlt I
• Arrays
• aIaI-1d

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Conversion to radix B

• Input N
• Output N in B-radix
• Algorithm
• K0
• Until N?0 do
• DigitKN mod B
• KK1
• NN div B
• End do

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Mathematical Induction

• First a puzzle with 2 solutions
• What monetary amounts can one not represent with
  2 and 5 cent coins?
• See the textbook for discussion
• Exercise What monetary amounts can one not
  represent with 2 and 5 cent coins?
• Answer 1 and 3 only
• Proofs
• Using remainder properties
• Using mathematical induction

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Questions?